On black hole thermodynamics and the entropy function formalism

TL;DR

This paper explores the relationship between entropy function and free energy in black hole thermodynamics, proposing a method to compute non-extremal black hole entropy from extremal cases, with concrete examples involving higher derivative corrections.

Contribution

It introduces a novel approach to estimate non-extremal black hole entropy by perturbing extremal geometries, extending the entropy function formalism to non-extremal cases.

Findings

Derived relation between entropy function and free energy.

Calculated entropy of non-extremal D1D5 and D1D5p branes with higher derivatives.

Provided explanation for the applicability of the entropy function method in higher derivative gravity.

Abstract

From the black hole thermodynamics point of view, we show that the entropy function $\mathbf{f}$ and the free energy $F$ are related via $\mathbf{f}=e_{I}q_{I}+\Omega_Hq_I{A_{\phi}^I}'-\frac{\partial F}{\partial r} |_{r_{H}}$. Assuming the entropy function is known for extremal black holes, we propose an approach to calculate the entropy of non-extremal cases by slightly moving the extremal black hole geometry from extremality. The entropy of non-extremal $D1D5$- and $D1D5p$-branes in the presence of higher derivative corrections are computed as concrete examples. An attempt has also been made to explain why the entropy function method can calculate the corrected entropy without knowing the exact form of black hole solution in higher derivative gravity theories.